A typical reconstruction limit of compressed sensing based on Lp-norm minimization

We consider the problem of reconstructing an $N$-dimensional continuous vector $\bx$ from $P$ constraints which are generated by its linear transformation under the assumption that the number of non-zero elements of $\bx$ is typically limited to $\rho N$ ($0\le \rho \le 1$). Problems of this type can be solved by minimizing a cost function with respect to the $L_p$-norm $||\bx||p=\lim{\epsilon \to +0}\sum_{i=1}^N |x_i|^{p+\epsilon}$, subject to the constraints under an appropriate condition. For several $p$, we assess a typical case limit $\alpha_c(\rho)$, which represents a critical relation between $\alpha=P/N$ and $\rho$ for successfully reconstructing the original vector by minimization for typical situations in the limit $N,P \to \infty$ with keeping $\alpha$ finite, utilizing the replica method. For $p=1$, $\alpha_c(\rho)$ is considerably smaller than its worst case counterpart, which has been rigorously derived by existing literature of information theory.

💡 Research Summary

The paper investigates the fundamental limits of reconstructing a high‑dimensional sparse vector from a set of linear measurements using $L_p$‑norm minimization. Let $\mathbf{x}\in\mathbb{R}^N$ be a continuous signal whose number of non‑zero components is limited to a fraction $\rho$ of the total dimension ($\rho N$ non‑zeros). A measurement matrix $A\in\mathbb{R}^{P\times N}$ with i.i.d. Gaussian entries generates $P$ linear constraints $\mathbf{y}=A\mathbf{x}$. The reconstruction problem is formulated as

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